We start by providing motivation for the unconstrained Jacobians problem introduced in the main text. We will continue our proof using contradiction. Figure 1: Fully-connected ResNet34 (Type 1 model) trained on MNIST.Figure 2: Fully-connected ResNet34 (Type 1 model) trained on FashionMNIST. Figure 10: Fully-connected ResNet34 (Type 1 model) trained on MNIST. Figure 24: Fully-connected ResNet34 (Type 1 model) trained on MNIST.

All UK children could be offered screening for type 1 diabetes using a simple finger-prick blood test, say researchers who have been running a large study. Currently, many young people go undiagnosed and risk developing a life-threatening complication called diabetic ketoacidosis that needs urgent hospital treatment. Identifying diabetes earlier could help avoid this and mean treatments to control problematic blood sugar levels can be given sooner. Some 17,000 children aged three to 13 have already been checked as part of the ELSA (Early Surveillance for Autoimmune diabetes) study, funded by diabetes charities. Imogen, who is 12 and from the West Midlands, is one of those found to have diabetes thanks to the screening.

However, there can be many different ways to do this, thus raising the problem of "

Type 2 diabetes prevention and treatment can benefit from personalized lifestyle prescriptions. However, the delivery of personalized lifestyle medicine prescriptions is limited by the shortage of trained professionals and the variability in physicians' expertise. We propose an offline contextual bandit approach that learns individualized lifestyle prescriptions from the aggregated NHANES profiles of 119,555 participants by minimizing the Magni glucose risk-reward function. The model encodes patient status and generates lifestyle medicine prescriptions, which are trained using a mixed-action Soft Actor-Critic algorithm. The task is treated as a single-step contextual bandit. The model is validated against lifestyle medicine prescriptions issued by three certified physicians from Xiangya Hospital. These results demonstrate that offline mixed-action SAC can generate risk-aware lifestyle medicine prescriptions from cross-sectional NHANES data, warranting prospective clinical validation.

From media platforms to chatbots, algorithms shape how people interact, learn, and discover information. Such interactions between users and an algorithm often unfold over multiple steps, during which strategic users can guide the algorithm to better align with their true interests by selectively engaging with content. However, users frequently exhibit inconsistent preferences: they may spend considerable time on content that offers little long-term value, inadvertently signaling that such content is desirable. Focusing on the user side, this raises a key question: what does it take for such users to align the algorithm with their true interests? To investigate these dynamics, we model the user's decision process as split between a rational system 2 that decides whether to engage and an impulsive system 1 that determines how long engagement lasts. We then study a multi-leader, single-follower extensive Stackelberg game, where users, specifically system 2, lead by committing to engagement strategies and the algorithm best-responds based on observed interactions. We define the burden of alignment as the minimum horizon over which users must optimize to effectively steer the algorithm. We show that a critical horizon exists: users who are sufficiently foresighted can achieve alignment, while those who are not are instead aligned to the algorithm's objective. This critical horizon can be long, imposing a substantial burden. However, even a small, costly signal (e.g., an extra click) can significantly reduce it. Overall, our framework explains how users with inconsistent preferences can align an engagement-driven algorithm with their interests in a Stackelberg equilibrium, highlighting both the challenges and potential remedies for achieving alignment.

C is the number of classes; and (RA4) top singular values of Residual Jacobians scale inversely with depth.